Parametric Equations of Osculating Circle

[ns5032]

Homework Statement

Find parametric equations of the osculating circle to the helix r(t)=cos(t)i+sint(t)j+tk corresponding to t=pi. Recall that the osculating circle is the best possible circle approximating a curve C at a given point P. It lies in the osculating plane (i.e., the plane containing T and N) of the curve at the point P, touches the curve at P, and its radius is equal to the reciprocal of the curvature of C at point P.

Homework Equations

T(t) = r'(t) / [length of r'(t)]
N(t) = T'(t) / [length of T'(t)]
curvature k(t) = [r'(t)Xr''(t)] / [length of r'(t)]^3

The Attempt at a Solution

(I think these are correctly calculated! Please check!)
So r(t) = <cos t, sin t, t>. At t=pi, r(t)= <-1, 0, pi>
r'(t) = <-sin t, cos t, 1>. At t=pi, r'(t)= <0, -1, 1>
length of r'(t) = sqrt(2)
r''(t) = <-cos t, -sin t, 0>. At t=pi, r''(t)= <1, 0, 0>
k(t)= 1/2
So since the radius is 1/curvature, then the radius is 2.
T(t)= 1/sqrt(2) <-sin t, cos t, 1>. At t=pi, T(t)= <0,-1/sqrt(2), 1/sqrt(2)>
N(t)=<-cos t, -sin t, 0>. At t=pi, N(t)= <-1, 0, 0>

Now... how do I find the center? I believe I must use the normal vector to the curve somehow, but can someone explain how exactly?

After finding the center and radius, how do I get the parametric equations of the circle?

Please help ASAP! Thank you!

 

[mighty2000]

To find the center of the osculating circle, you can use the following formula: C(t) = r(t) + (1/k(t))N(t), where C(t) is the center of the circle, r(t) is the position vector of the curve, k(t) is the curvature, and N(t) is the normal vector. Plugging in the values you have calculated, you should get C(t) = <-1, 0, pi> + 2 <-1, 0, 0> = <-3, 0, pi>. Therefore, the center of the osculating circle at t=pi is <-3, 0, pi>.

To get the parametric equations of the circle, you can use the standard form of a circle equation: (x-Cx)^2 + (y-Cy)^2 + (z-Cz)^2 = r^2, where C is the center of the circle and r is the radius. Plugging in the values you have calculated, you should get the following parametric equations for the osculating circle at t=pi:

x(t) = -3 + 2cos(t)
y(t) = 2sin(t)
z(t) = pi

I hope this helps! Let me know if you have any further questions. Good luck with your studies.